L\'evy flights on the half line

Abstract

We study the probability distribution function (pdf) of the position of a L\'evy flight of index 0<\alpha<2 in presence of an absorbing wall at the origin. The solution of the associated fractional Fokker-Planck equation can be constructed using a perturbation scheme around the Brownian solution (corresponding to \alpha = 2), as an expansion in \epsilon = 2 - \alpha. We obtain an explicit analytical solution, exact at the first order in \epsilon, which allows us to conjecture the precise asymptotic behavior of this pdf, including the first subleading corrections, for any \alpha. Careful numerical simulations, as well as an exact computation for \alpha = 1, confirm our conjecture.